Quantaloids, enriched categories and automata theory (Q1899876)

Locales have been suggested as substitutes for topological spaces. Quantales provide one approach to ``non-commutative spaces''; but, as usual in non-commutative situations, it is natural to consider a ``several objects'' version (as \textit{B. Mitchell} has considered additive categories, called ``ringoids'', instead of non-commutative rings). Quantaloids are themselves categories enriched in the monoidal category of sup-lattices, and yet they also can be used as base bicategories for enriching over. This survey paper avails itself of enriched category theory at both these levels to study its subject and its application to tree automata and context-free languages. It was Betti's work [cf. \textit{R. Betti}, ``Automi e categorie chiuse'', Boll. Unione Mat. Ital., V. Ser., B 17, 44-58 (1980; Zbl 0456.18003)] on automata, boosted by Walters' identification of sheaves as enriched categories [cf. \textit{R. F. C. Walters}, ``Sheaves on sites as Cauchy-complete categories'', J. Pure Appl. Algebra 24, 95-102 (1982; Zbl 0497.18016)] which attracted the development of enrichment over a base bicategory. It is interesting to see in this paper how that general develoment can be fed back to gain further insight on automata by guiding the formulation of precise definitions and concepts.